{
  "dimension_window": [
    0,
    1,
    2,
    3,
    4
  ],
  "forward_pole_orders": [
    1,
    1,
    1,
    1,
    1
  ],
  "opposite_pole_orders": [
    1,
    2,
    3,
    4,
    5
  ],
  "normal_residue_sample": {
    "N": 3,
    "A": "3",
    "residue": "1/27",
    "pole_orders": [
      [
        1,
        3
      ],
      [
        3,
        1
      ]
    ],
    "positive_orientation": true,
    "circle_picture": "Schematic: the displayed circle surrounds only zero; the proof permits every radius 0<r<|A|."
  },
  "spatial_period_sample": {
    "d": 1,
    "epsilon": "1",
    "a": "3",
    "exact_rational_values": [
      {
        "b": "-2",
        "period": "2/5",
        "bound": "2"
      },
      {
        "b": "-79/40",
        "period": "3200/8159",
        "bound": "3200/1681"
      },
      {
        "b": "-39/20",
        "period": "800/2079",
        "bound": "800/441"
      },
      {
        "b": "-77/40",
        "period": "3200/8471",
        "bound": "3200/1849"
      },
      {
        "b": "-19/10",
        "period": "200/539",
        "bound": "200/121"
      },
      {
        "b": "-15/8",
        "period": "128/351",
        "bound": "128/81"
      },
      {
        "b": "-37/20",
        "period": "800/2231",
        "bound": "800/529"
      },
      {
        "b": "-73/40",
        "period": "3200/9071",
        "bound": "3200/2209"
      },
      {
        "b": "-9/5",
        "period": "25/72",
        "bound": "25/18"
      },
      {
        "b": "-71/40",
        "period": "3200/9359",
        "bound": "3200/2401"
      },
      {
        "b": "-7/4",
        "period": "32/95",
        "bound": "32/25"
      },
      {
        "b": "-69/40",
        "period": "3200/9639",
        "bound": "3200/2601"
      },
      {
        "b": "-17/10",
        "period": "200/611",
        "bound": "200/169"
      },
      {
        "b": "-67/40",
        "period": "3200/9911",
        "bound": "3200/2809"
      },
      {
        "b": "-33/20",
        "period": "800/2511",
        "bound": "800/729"
      },
      {
        "b": "-13/8",
        "period": "128/407",
        "bound": "128/121"
      },
      {
        "b": "-8/5",
        "period": "50/161",
        "bound": "50/49"
      },
      {
        "b": "-63/40",
        "period": "3200/10431",
        "bound": "3200/3249"
      },
      {
        "b": "-31/20",
        "period": "800/2639",
        "bound": "800/841"
      },
      {
        "b": "-61/40",
        "period": "3200/10679",
        "bound": "3200/3481"
      },
      {
        "b": "-3/2",
        "period": "8/27",
        "bound": "8/9"
      },
      {
        "b": "-59/40",
        "period": "3200/10919",
        "bound": "3200/3721"
      },
      {
        "b": "-29/20",
        "period": "800/2759",
        "bound": "800/961"
      },
      {
        "b": "-57/40",
        "period": "3200/11151",
        "bound": "3200/3969"
      },
      {
        "b": "-7/5",
        "period": "25/88",
        "bound": "25/32"
      },
      {
        "b": "-11/8",
        "period": "128/455",
        "bound": "128/169"
      },
      {
        "b": "-27/20",
        "period": "800/2871",
        "bound": "800/1089"
      },
      {
        "b": "-53/40",
        "period": "3200/11591",
        "bound": "3200/4489"
      },
      {
        "b": "-13/10",
        "period": "200/731",
        "bound": "200/289"
      },
      {
        "b": "-51/40",
        "period": "3200/11799",
        "bound": "3200/4761"
      },
      {
        "b": "-5/4",
        "period": "32/119",
        "bound": "32/49"
      },
      {
        "b": "-49/40",
        "period": "3200/11999",
        "bound": "3200/5041"
      },
      {
        "b": "-6/5",
        "period": "50/189",
        "bound": "50/81"
      },
      {
        "b": "-47/40",
        "period": "3200/12191",
        "bound": "3200/5329"
      },
      {
        "b": "-23/20",
        "period": "800/3071",
        "bound": "800/1369"
      },
      {
        "b": "-9/8",
        "period": "128/495",
        "bound": "128/225"
      },
      {
        "b": "-11/10",
        "period": "200/779",
        "bound": "200/361"
      },
      {
        "b": "-43/40",
        "period": "3200/12551",
        "bound": "3200/5929"
      },
      {
        "b": "-21/20",
        "period": "800/3159",
        "bound": "800/1521"
      },
      {
        "b": "-41/40",
        "period": "3200/12719",
        "bound": "3200/6241"
      },
      {
        "b": "-1",
        "period": "1/4",
        "bound": "1/2"
      },
      {
        "b": "-39/40",
        "period": "3200/12879",
        "bound": "3200/6561"
      },
      {
        "b": "-19/20",
        "period": "800/3239",
        "bound": "800/1681"
      },
      {
        "b": "-37/40",
        "period": "3200/13031",
        "bound": "3200/6889"
      },
      {
        "b": "-9/10",
        "period": "200/819",
        "bound": "200/441"
      },
      {
        "b": "-7/8",
        "period": "128/527",
        "bound": "128/289"
      },
      {
        "b": "-17/20",
        "period": "800/3311",
        "bound": "800/1849"
      },
      {
        "b": "-33/40",
        "period": "3200/13311",
        "bound": "3200/7569"
      },
      {
        "b": "-4/5",
        "period": "50/209",
        "bound": "50/121"
      },
      {
        "b": "-31/40",
        "period": "3200/13439",
        "bound": "3200/7921"
      },
      {
        "b": "-3/4",
        "period": "32/135",
        "bound": "32/81"
      },
      {
        "b": "-29/40",
        "period": "3200/13559",
        "bound": "3200/8281"
      },
      {
        "b": "-7/10",
        "period": "200/851",
        "bound": "200/529"
      },
      {
        "b": "-27/40",
        "period": "3200/13671",
        "bound": "3200/8649"
      },
      {
        "b": "-13/20",
        "period": "800/3431",
        "bound": "800/2209"
      },
      {
        "b": "-5/8",
        "period": "128/551",
        "bound": "128/361"
      },
      {
        "b": "-3/5",
        "period": "25/108",
        "bound": "25/72"
      },
      {
        "b": "-23/40",
        "period": "3200/13871",
        "bound": "3200/9409"
      },
      {
        "b": "-11/20",
        "period": "800/3479",
        "bound": "800/2401"
      },
      {
        "b": "-21/40",
        "period": "3200/13959",
        "bound": "3200/9801"
      },
      {
        "b": "-1/2",
        "period": "8/35",
        "bound": "8/25"
      },
      {
        "b": "-19/40",
        "period": "3200/14039",
        "bound": "3200/10201"
      },
      {
        "b": "-9/20",
        "period": "800/3519",
        "bound": "800/2601"
      },
      {
        "b": "-17/40",
        "period": "3200/14111",
        "bound": "3200/10609"
      },
      {
        "b": "-2/5",
        "period": "50/221",
        "bound": "50/169"
      },
      {
        "b": "-3/8",
        "period": "128/567",
        "bound": "128/441"
      },
      {
        "b": "-7/20",
        "period": "800/3551",
        "bound": "800/2809"
      },
      {
        "b": "-13/40",
        "period": "3200/14231",
        "bound": "3200/11449"
      },
      {
        "b": "-3/10",
        "period": "200/891",
        "bound": "200/729"
      },
      {
        "b": "-11/40",
        "period": "3200/14279",
        "bound": "3200/11881"
      },
      {
        "b": "-1/4",
        "period": "32/143",
        "bound": "32/121"
      },
      {
        "b": "-9/40",
        "period": "3200/14319",
        "bound": "3200/12321"
      },
      {
        "b": "-1/5",
        "period": "25/112",
        "bound": "25/98"
      },
      {
        "b": "-7/40",
        "period": "3200/14351",
        "bound": "3200/12769"
      },
      {
        "b": "-3/20",
        "period": "800/3591",
        "bound": "800/3249"
      },
      {
        "b": "-1/8",
        "period": "128/575",
        "bound": "128/529"
      },
      {
        "b": "-1/10",
        "period": "200/899",
        "bound": "200/841"
      },
      {
        "b": "-3/40",
        "period": "3200/14391",
        "bound": "3200/13689"
      },
      {
        "b": "-1/20",
        "period": "800/3599",
        "bound": "800/3481"
      },
      {
        "b": "-1/40",
        "period": "3200/14399",
        "bound": "3200/14161"
      },
      {
        "b": "0",
        "period": "2/9",
        "bound": "2/9"
      },
      {
        "b": "1/40",
        "period": "3200/14399",
        "bound": "3200/14161"
      },
      {
        "b": "1/20",
        "period": "800/3599",
        "bound": "800/3481"
      },
      {
        "b": "3/40",
        "period": "3200/14391",
        "bound": "3200/13689"
      },
      {
        "b": "1/10",
        "period": "200/899",
        "bound": "200/841"
      },
      {
        "b": "1/8",
        "period": "128/575",
        "bound": "128/529"
      },
      {
        "b": "3/20",
        "period": "800/3591",
        "bound": "800/3249"
      },
      {
        "b": "7/40",
        "period": "3200/14351",
        "bound": "3200/12769"
      },
      {
        "b": "1/5",
        "period": "25/112",
        "bound": "25/98"
      },
      {
        "b": "9/40",
        "period": "3200/14319",
        "bound": "3200/12321"
      },
      {
        "b": "1/4",
        "period": "32/143",
        "bound": "32/121"
      },
      {
        "b": "11/40",
        "period": "3200/14279",
        "bound": "3200/11881"
      },
      {
        "b": "3/10",
        "period": "200/891",
        "bound": "200/729"
      },
      {
        "b": "13/40",
        "period": "3200/14231",
        "bound": "3200/11449"
      },
      {
        "b": "7/20",
        "period": "800/3551",
        "bound": "800/2809"
      },
      {
        "b": "3/8",
        "period": "128/567",
        "bound": "128/441"
      },
      {
        "b": "2/5",
        "period": "50/221",
        "bound": "50/169"
      },
      {
        "b": "17/40",
        "period": "3200/14111",
        "bound": "3200/10609"
      },
      {
        "b": "9/20",
        "period": "800/3519",
        "bound": "800/2601"
      },
      {
        "b": "19/40",
        "period": "3200/14039",
        "bound": "3200/10201"
      },
      {
        "b": "1/2",
        "period": "8/35",
        "bound": "8/25"
      },
      {
        "b": "21/40",
        "period": "3200/13959",
        "bound": "3200/9801"
      },
      {
        "b": "11/20",
        "period": "800/3479",
        "bound": "800/2401"
      },
      {
        "b": "23/40",
        "period": "3200/13871",
        "bound": "3200/9409"
      },
      {
        "b": "3/5",
        "period": "25/108",
        "bound": "25/72"
      },
      {
        "b": "5/8",
        "period": "128/551",
        "bound": "128/361"
      },
      {
        "b": "13/20",
        "period": "800/3431",
        "bound": "800/2209"
      },
      {
        "b": "27/40",
        "period": "3200/13671",
        "bound": "3200/8649"
      },
      {
        "b": "7/10",
        "period": "200/851",
        "bound": "200/529"
      },
      {
        "b": "29/40",
        "period": "3200/13559",
        "bound": "3200/8281"
      },
      {
        "b": "3/4",
        "period": "32/135",
        "bound": "32/81"
      },
      {
        "b": "31/40",
        "period": "3200/13439",
        "bound": "3200/7921"
      },
      {
        "b": "4/5",
        "period": "50/209",
        "bound": "50/121"
      },
      {
        "b": "33/40",
        "period": "3200/13311",
        "bound": "3200/7569"
      },
      {
        "b": "17/20",
        "period": "800/3311",
        "bound": "800/1849"
      },
      {
        "b": "7/8",
        "period": "128/527",
        "bound": "128/289"
      },
      {
        "b": "9/10",
        "period": "200/819",
        "bound": "200/441"
      },
      {
        "b": "37/40",
        "period": "3200/13031",
        "bound": "3200/6889"
      },
      {
        "b": "19/20",
        "period": "800/3239",
        "bound": "800/1681"
      },
      {
        "b": "39/40",
        "period": "3200/12879",
        "bound": "3200/6561"
      },
      {
        "b": "1",
        "period": "1/4",
        "bound": "1/2"
      },
      {
        "b": "41/40",
        "period": "3200/12719",
        "bound": "3200/6241"
      },
      {
        "b": "21/20",
        "period": "800/3159",
        "bound": "800/1521"
      },
      {
        "b": "43/40",
        "period": "3200/12551",
        "bound": "3200/5929"
      },
      {
        "b": "11/10",
        "period": "200/779",
        "bound": "200/361"
      },
      {
        "b": "9/8",
        "period": "128/495",
        "bound": "128/225"
      },
      {
        "b": "23/20",
        "period": "800/3071",
        "bound": "800/1369"
      },
      {
        "b": "47/40",
        "period": "3200/12191",
        "bound": "3200/5329"
      },
      {
        "b": "6/5",
        "period": "50/189",
        "bound": "50/81"
      },
      {
        "b": "49/40",
        "period": "3200/11999",
        "bound": "3200/5041"
      },
      {
        "b": "5/4",
        "period": "32/119",
        "bound": "32/49"
      },
      {
        "b": "51/40",
        "period": "3200/11799",
        "bound": "3200/4761"
      },
      {
        "b": "13/10",
        "period": "200/731",
        "bound": "200/289"
      },
      {
        "b": "53/40",
        "period": "3200/11591",
        "bound": "3200/4489"
      },
      {
        "b": "27/20",
        "period": "800/2871",
        "bound": "800/1089"
      },
      {
        "b": "11/8",
        "period": "128/455",
        "bound": "128/169"
      },
      {
        "b": "7/5",
        "period": "25/88",
        "bound": "25/32"
      },
      {
        "b": "57/40",
        "period": "3200/11151",
        "bound": "3200/3969"
      },
      {
        "b": "29/20",
        "period": "800/2759",
        "bound": "800/961"
      },
      {
        "b": "59/40",
        "period": "3200/10919",
        "bound": "3200/3721"
      },
      {
        "b": "3/2",
        "period": "8/27",
        "bound": "8/9"
      },
      {
        "b": "61/40",
        "period": "3200/10679",
        "bound": "3200/3481"
      },
      {
        "b": "31/20",
        "period": "800/2639",
        "bound": "800/841"
      },
      {
        "b": "63/40",
        "period": "3200/10431",
        "bound": "3200/3249"
      },
      {
        "b": "8/5",
        "period": "50/161",
        "bound": "50/49"
      },
      {
        "b": "13/8",
        "period": "128/407",
        "bound": "128/121"
      },
      {
        "b": "33/20",
        "period": "800/2511",
        "bound": "800/729"
      },
      {
        "b": "67/40",
        "period": "3200/9911",
        "bound": "3200/2809"
      },
      {
        "b": "17/10",
        "period": "200/611",
        "bound": "200/169"
      },
      {
        "b": "69/40",
        "period": "3200/9639",
        "bound": "3200/2601"
      },
      {
        "b": "7/4",
        "period": "32/95",
        "bound": "32/25"
      },
      {
        "b": "71/40",
        "period": "3200/9359",
        "bound": "3200/2401"
      },
      {
        "b": "9/5",
        "period": "25/72",
        "bound": "25/18"
      },
      {
        "b": "73/40",
        "period": "3200/9071",
        "bound": "3200/2209"
      },
      {
        "b": "37/20",
        "period": "800/2231",
        "bound": "800/529"
      },
      {
        "b": "15/8",
        "period": "128/351",
        "bound": "128/81"
      },
      {
        "b": "19/10",
        "period": "200/539",
        "bound": "200/121"
      },
      {
        "b": "77/40",
        "period": "3200/8471",
        "bound": "3200/1849"
      },
      {
        "b": "39/20",
        "period": "800/2079",
        "bound": "800/441"
      },
      {
        "b": "79/40",
        "period": "3200/8159",
        "bound": "3200/1681"
      },
      {
        "b": "2",
        "period": "2/5",
        "bound": "2"
      }
    ]
  },
  "proof_locators": [
    "OI.6\u2013OI.15",
    "OI.18\u2013OI.20",
    "OI.21\u2013OI.25"
  ],
  "analytic_inverse_product_claimed": false,
  "free_context": "https://arxiv.org/abs/2102.08861v2"
}